Expand all Collapse all | Results 1 - 9 of 9 |
1. CJM Online first
Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$ We prove the congruence relation for the mod-$p$ reduction of Shimura
varieties associated to a unitary similitude group $GU(n-1,1)$ over
$\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$
is even was obtained by T. Wedhorn and O. B?ltel, as a special case
of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny
space is dense. This condition fails in our case. We show that every
supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$
is annihilated by a degree one polynomial in the Frobenius element
$F$, which implies the congruence relation.
Keywords:Shimura varieties, congruence relation Categories:11G18, 14G35, 14K10 |
2. CJM Online first
Twists of Shimura Curves Consider a Shimura curve $X^D_0(N)$ over the rational
numbers. We determine criteria for the twist by an Atkin-Lehner
involution to have points over a local field. As a corollary we give a
new proof of the theorem of Jordan-LivnÃ© on $\mathbf{Q}_p$ points
when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$
points when $p\mid N$. We also give congruence conditions for roots
modulo $p$ of Hilbert class polynomials.
Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves Categories:11G18, 14G35, 11G15, 11G10 |
3. CJM 2013 (vol 65 pp. 1125)
On the Global Structure of Special Cycles on Unitary Shimura Varieties In this paper, we study the reduced loci of special cycles on local
models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits
theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.
Keywords:Shimura varieties, local models, special cycles Category:14G35 |
4. CJM Online first
Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves The purpose of this note is introducing a method for proving the
existence of no rational points on a coarse moduli space $X$ of abelian varieties
over a given number field $K$, in cases where the moduli problem is not fine and
points in $X(K)$ may not be represented by an abelian variety (with additional structure)
admitting a model over the field $K$. This is typically the case when the abelian
varieties that are being classified have even dimension. The main idea, inspired on
the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a
point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still
attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient
group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided
$\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$.
We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary
quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.
Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstruction Categories:11G18, 14G35, 14G05 |
5. CJM 2011 (vol 64 pp. 1248)
Darmon's Points and Quaternionic Shimura Varieties In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's
points.
Keywords:elliptic curves, Stark-Heegner points, quaternionic Shimura varieties Categories:11G05, 14G35, 11F67, 11G40 |
6. CJM 2010 (vol 62 pp. 668)
The Supersingular Locus of the Shimura Variety for GU(1,s) In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using DieudonnÃ© theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.
Categories:14G35, 11G18, 14K10 |
7. CJM 2009 (vol 61 pp. 828)
Twisted Gross--Zagier Theorems The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate
heights of complex multiplication points on the modular curve $X_0(N)$
(and on Shimura curve analogues) with the central derivatives of
automorphic $L$-function. We extend these results to include certain
CM points on modular curves of the form
$X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues).
These results are motivated by applications to Hida theory
that can be found in the companion article
"Central derivatives of $L$-functions in Hida families", Math.\ Ann.\
\textbf{399}(2007), 803--818.
Categories:11G18, 14G35 |
8. CJM 2008 (vol 60 pp. 734)
Genus 2 Curves with Quaternionic Multiplication We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our $j$-functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using $j$, we construct the fields of moduli and definition
for some moduli problems associated to the Atkin--Lehner group
actions.
Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli Categories:11G18, 14G35 |
9. CJM 2002 (vol 54 pp. 352)
On Connected Components of Shimura Varieties We study the cohomology of connected components of Shimura varieties
$S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on
the stabilization of the twisted trace formula, due to Kottwitz and
Shelstad. More precisely, for each character $\olomega$ on
the group of connected components of $S_{K^p}$ we define an operator
$L(\omega)$ on the cohomology groups with compact supports $H^i_c
(S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual
trace of the composition of $L(\omega)$ with a Hecke operator $f$ away
from $p$ and a sufficiently high power of a geometric Frobenius
$\Phi^r_p$, can be expressed as a sum of $\omega$-{\em weighted}
(twisted) orbital integrals (where $\omega$-{\em weighted} means that
the orbital integrals and twisted orbital integrals occuring here each
have a weighting factor coming from the character $\olomega$).
As the crucial step, we define and study a new invariant $\alpha_1
(\gamma_0; \gamma, \delta)$ which is a refinement of the invariant
$\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done
by using a theorem of Reimann and Zink.
Categories:14G35, 11F70 |